After decisive votes in the House and Senate, Congress is on the brink of a rare thing: bipartisan support for a piece of President Barack Obama’s jobs bill.

And Senate Majority Leader Harry Reid isn’t letting the moment go without attempting to make one good bipartisan turn yield another.

The Senate voted overwhelmingly (94 to 1) Monday evening to move forward on a bill that would repeal a 3 percent withholding tax on federal and state government contractors, a regulation of George W. Bush’s presidency that hasn’t taken effect yet. It’s slated to, in 2013 — that is, unless Congress repeals it first, which it appears to be on its way to: the House voted 405 to 16 in favor of the repeal last month.

Reid now plans to attach language that would give a tax break to companies who hire recent veterans — another bit of Obama’s jobs plan — to the tax withholding bill.

“It is time for this country to make good on its promise,” Reid said on the Senate floor Monday afternoon, citing veterans’ unemployment statistics: 12.1 percent among those who have served since September 11, and 22 percent for those under age 25.

“As we pay tribute this week to the millions of Americans veterans who have faithfully served our flag, Democrats will introduce legislation to put those men and women back to work,” Reid said.

If all goes according to plan, the bill would clear the Senate just in time for Veterans’ Day.

But that’s not likely the end of the process.

The House didn’t vote on a bill that includes the veterans hiring tax incentive, and those lawmakers aren’t in Washington this week.

Republican leaders in the House aren’t opposed to the idea, but they’d rather see a bill of their own move forward instead of what’s in the president’s jobs package.

That bill is the so-called Veterans Opportunity to Work Act, which would provide job retraining services to veterans, especially the long-term unemployed. It passed the House last month 418 to 6.

The provision the Senate aims to take up also addresses retraining, but goes further with the tax incentives: up to $5,600 per veteran and $9,600 per disabled veteran that companies hire. The program is geared, in particular, toward veterans who have been unemployed for six months or more.

Reid all but dared Republicans to vote against it.

“Senate Republicans unanimously opposed our last three jobs bills, although those bills had the support of a vast majority of Americans...Their obstruction has cost hundreds of thousands of teacher and first responder jobs. It has cost hundreds of thousands of construction jobs, and put reconstruction of our nation’s crumbling roads, bridges and runways on hold,” he said. “Now we will see whether Senate Republicans are willing to put jobs for veterans at risk as well.”

Republican Leader Mitch McConnell didn’t speak specifically about veterans on the Senate floor Monday, but scolded Reid for standing in the way of what bipartisan cooperation could actually achieve.

“The problem is, every time Republicans pass one of these bills over in the House, Democrats here in the Senate refuse to take it up. The Democrats who run the Senate are just letting all these bills die,” McConnell said.

While the Senate hashes it out, Obama moved ahead with a roster of executive orders today meant to address veterans’ employment issues. They are as follows:

• The “Veteran Gold Card”: available to all post-9/11 veterans, holders can access enhanced One-Stop career center services, including six months of personal case management and counseling.

• The “My Next Move for Veterans” initiative: an online database that lets former servicemembers find civilian jobs that fit their military skills, simply by entering their military occupation code.

• A Veterans’ Jobs Bank: a searchable jobs listings tool; employers who choose to list their jobs on it expected to have an interest in hiring former members of the military.